Zero-coupon bond options

Cox, Ingersoll and Ross [22] and Jamshidian [42] demonstrate that closed-form solutions for zero-coupon bond options can be derived for single-factor square root and Gaussian models. More generally, Duffie, Pan and Singleton... 

In chapter (2), we derive a unified framework for the computation of the price of an option on a zero-coupon bond and a coupon bond by applying the well known Fourier inversion scheme. Therefore, we introduce the transform t (z), which later on can be seen as a characteristic function. In case of zero-coupon bond options we are able to find a closed-form solution for the transform t z) and apply standard Fourier inversion techniques. Unfortunately, assuming a multi-factor framework there exists no closed-form solution of the characteristic function Et z) given a coupon bond option. Hence, in this case Fourier inversion techniques fail. 

The price process under the new measure Tq, either is used to derive the formula for the zero-coupon bond option (see section (5.2.1)), the characteristic function in (5.2.2), or finally to compute the moments of the underlying random variable (section (5.3.3) and (5.3.4)). 

Starting from the risk-neutral bond price dynamics (5.4), we derive the well known closed-form solution for the price of a zero-coupon bond option. Thus, as shown in section (2.1) the price of a call option on a discount bond is given by... 

Following the last section, we introduce the FRFT technique to derive the price of a zero-coupon bond option. In doing so, we are able to compare the option price coming from the FRFT approach with the appropriate closed-form solution (5.27). 

In section (5.2.1), we have derived the closed-form solution for the price of a zero-coupon bond option. Then, later on in section (5.2.2) we introduced the FRFT-technique and showed that this method works excellent for a wide range of strike prices by solving the Fourier inversion numerically. Now, we show that also the IFF is an efficient and accurate method to compute the single exercise probabilities (see section (2.2)) for a G 0,1 via... 

Following chapter (5.2) we obtain the price of a zero-coupon bond option by computing the risk-neutral probabilities... 

The zero-correlation (y = ) price of a coupon bond option with a moneyness 1.14 is about 1.7 times as high as the corresponding option price obtained by a perfect correlation stnjcture (y = 0). The corresponding zero-coupon bond option price is about 60 times as high as its perfect correlation equivalent. 

In this thesis we derived new methods for the pricing of fixed income derivatives, especially for zero-coupon bond options (caps/floor) and coupon bond options (swaptions). These options are the most widely traded interest rate derivatives. In general caps/floors can be seen as a portfolio of zero-coupon bond options, whereas a swaption effectively equals an option on a coupon bond (see chapter (2)). The market of these LIBOR-based interest rate derivatives is tremendous (more than 10 trillion USD in notional value) and therefore accurate and efficient pricing methods are of enormous practical importance. 

---